F Fermi National Accelerator Laboratory
FERMILAB-FN-684
Short-Bunch Production and Microwave Instability Near Transition
K.Y. Ng and J. Norem
Fermi National Accelerator LaboratoryP.O. Box 500, Batavia, Illinois 60510
October 1999
Operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000 with the United States Department of Energy
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Short-Bunch Production and MicrowaveInstability Near Transition
K.Y. Ng1 and J. Norem2
1Fermi National Accelerator Laboratory,3 P.O. Box 500, Batavia, IL 605102HEP Division, Argonne National Laboratory, Argonne, IL 60439
Abstract. Some methods of making short bunches are reviewed. The experiment per-formed at the Brookhaven AGS for bunching near transition is reported. Microwaveinstability for coasting beam and bunched beam near transition is discussed and sim-ulations are presented.
I INTRODUCTION
For the proton driver of the muon collider, bunching of intense proton bunchesto rms length στ ≤ 2 ns at extraction is desirable. There are two primary reasons.First, the proton bunch length is the only piece of information transmitted to thepions produced in the target and muons resulting in the decay of the pions. Theshorter the length of the proton bunches, the less cooling of the muons will be neces-sary. Second, it will be easier to separate the muons polarized in the two helicities.The shorter the proton bunch length will result in higher muon polarization. Thefollowing are some ways to achieve narrow bunches using rf gymnastics:
(1) Lowering and increasing rf voltage
The rf is reduced adiabatically until the bunch spreads out and fills the bucket.The rf voltage is raised again suddenly. In a quarter synchrotron oscillation, a nar-row bunch is obtained. The adiabatic process may take very long in order to allowthe bunch to follow the change in the bucket. However, for a high-intensity bunchto stay at low momentum spread for a long time, it is likely that the microwaveinstability will develop. In order to avoid instability, we can snap the rf voltagedown suddenly so that the rf bucket changes from Fig. 1(a) to 1(b). The bunchwill be lengthened after a quarter synchrotron oscillation. The rf voltage is thensnapped up again as in Fig. 1(c) and finally the lengthened bunch rotates into anarrow bunch. Of course, the rf nonlinearity will show up during bunch rotations.A second or third-order harmonic cavity will help in cancelling the rf nonlinearity.In practice, this method can shorten the bunch by a factor of at most 3 to 4.
(2) Debunching at unstable fixed point
The rf phase is suddenly shifted by 180 so that the bunch originally centered atthe stable fixed point in Fig. 2(a) finds itself centered at the unstable fixed pointin Fig. 2(b). The bunch will therefore spread out along the separatrices. After awhile, the rf phase is shifted back by 180 as in Fig. 2(c). Synchrotron oscillationbetween 1
4and 1
2period will rotate the bunch into a narrow one. Again nonlinearity
of the rf will show up in the bunch shape and a higher-order harmonic cavity will
3) Operated by the Universities Research Association, Inc., under contract with the U.S. Depart-ment of Energy.
c© 1995 American Institute of Physics 1
2
help. Also, this process may be slow because movement along the separatrices isslow. In practice, this method can shorten the bunch by a factor of at most 3 to 4.
(c)(a) (b)
FIGURE 1. Bunch shortening is performed by snapping down the rf voltage Vrf , rotating for 14
synchrotron oscillation, snapping up Vrf , and rotating for another 14 synchrotron oscillation.
(a) (b) (c)FIGURE 2. Bunch shortening is performed by shifting the rf phase by 180, allowing thebunch to spread along the separatrices, shifting the rf phase by −180, and rotating for 1
4 to 12
synchrotron oscillation.
(3) Rebunching at higher frequency
During the end of the ramping, the frequency of the rf system is jumped to thenext higher multiple of the circulating frequency. This process continues and thebunch will gradually be shortened by following the change in width of the bucket.In practice, the rf frequency of a rf system cannot be changed by very much. There-fore, there must be several rf systems with frequencies one above the other, so thatthe lower-frequency system will be replaced by the next higher one, etc. Thus,this method involve several high-frequency and high-voltage rf systems and will beexpensive. Also the whole procedure will be slow.
(4) Bunch-shortening near transition
At or near transition, there is little or no phase motion of the bunch particles.Thus, the particles continue to gain or lose energy according to the rf voltage theysee, as is illustrated in Fig. 3(a), where the phase axis represents the rf phase of theparticle when crossing the rf cavity gap. The bunch will shear in the momentumspread direction. A partial rotation will produce a narrow bunch, as depicted inFig. 3(b). It is desirable to make the final bunching as fast as possible because ofthe large instantaneous currents produced in the ring, which can drive a variety
3
of instabilities. The final bunching can be made quite fast if the transition energycan be moved farther away from the beam energy and/or the rf voltage can beraised during the final rotation, thus raising the synchrotron frequency just beforeextraction. One of the merits of this method is that no additional hardware, such ashigher-harmonic cavities, is required. Also this method uses only the linear part ofthe rf wave and a small synchrotron phase rotation. Thus the rotation can be madequite linear. We do not need to operate the ring at or near transition all the time.With the flexible momentum-compaction (FMC) lattice, the transition gamma canbe varied to a large extent by varying the gradients of a pair of quadrupoles [1].An example is shown in Fig. 4, where each of the two F-quadrupoles at a distanceabout 1
3from the entrance and exit of the FMC module has been split into a pair
denoted by QFS and QF2. By varying the gradients of the QFS and QF2, thelarge variation of transition gamma is shown in the left plot of Fig. 5, and thecorresponding values of the momentum-compaction factor α are listed on the right.The betatron tunes have been kept nearly unchanged during the variation. As a
rf
(a)
rf
(b)FIGURE 3. Bunch shortening is performed by allowing the bunch to shear very near to tran-sition in the momentum direction, raising the rf, and rotating for < 1
4 synchrotron oscillation.
s in mFIGURE 4. Two identical pairs of F-quadrupoles QFS and QF2 are installed at about 1
3from
the entrance and exit of the FMC module. γt can be varied by varying their gradients.
4
QFS (T/m) QF2 (T/m) α
12.89 38.60 +0.00042417.23 35.56 +0.00019119.52 33.56 +0.00002120.29 32.77 +0.00000127.23 27.08 −0.00000027.28 27.04 −0.00000327.50 26.85 −0.00001532.93 21.93 −0.000335
FIGURE 5. When the gradients of the quadrupole pairs are varied, the transition gamma (left)and the corresponding momentum-compaction factor α (right) change by a wide range.
result, we can move close to or right at transition only at the moment when wewant to make short bunches.
II EXPERIMENT ON BUNCH-SHORTENING NEARTRANSITION
An experiment was performed at the Brookhaven AGS to demonstrate bunch-shortening near transition [2]. The operating mode of the AGS is shown in the leftplot of Fig. 6. The maximum beam energy was reduced by flat-topping at 7 GeV,which shortened the acceleration period. The γt-jump system was modified to givea short flat-top period before the transition energy dropped. The beam was flat-
0
1
2
3
4
5
6
7
8
9
10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.
γt
γ
Np x 1012
IHsdB/dt
A (Vsec)
10 ns
Time (s) ←10 ns→FIGURE 6. Left: The operating mode of the AGS, showing the magnet ramp dB/dt in10−1 T/s, beam gamma γ, transition gamma γt, sextupole current IHs in 10 A, and longitu-dinal bunch area A in eV-s. Right: Mountain range plots of the bunch for about 50 ms starting10 ms before the transition energy was dropped to near the beam energy.
5
topped for 300 ms before the magnet guide field was raised slightly and then rampeddown. Only one bunch was injected, which has a bunch area of 1.5± 0.05 eV-s andan intensity of 3 to 5× 1012 protons.
Because of the AGS γt-jump mechanism, the energy of the beam was kept atmore than one unit of γ below γt. At about 0.35 s after injection, γt was droppedso that the beam was close to transition, with |γt−γ| < 0.05. The beam started toshear and at the same time rotate slowly. Here, no special hardware was available tomove the beam away from transition and no other higher rf voltage was providedto perform the final partial rotation depicted in Fig. 3(b). For this reason, thebeam cannot be too close to γt, otherwise the partial rotation will take very long.Likewise, it cannot be too far from γt, otherwise the bunch will shear not only inthe momentum direction as required in Fig. 3(a), but also in the phase direction,so that a tall and narrow beam will not be produced. During this run, a newmeasurement of γt = 8.34±0.05 was made, which was best determined by measuringthe synchrotron frequency.
A sample result is shown in the right plot of Fig. 6, which consists of mountain-range plots of the peak beam current and instantaneous current versus the machinephase sampled by the wall-gap monitor from 10 ms before the γt dropped to 40 msafter. We can see obviously that the bunch became narrow after the transitionenergy was dropped to near the beam energy.
The beam current versus time during the final bunch rotation is shown in the leftplot of Fig. 7. The bunch shape corresponding to the situation when it is shortestwas shown in the right plot of Fig. 7, together with its shape before the transitionenergy was lowered. The shortest rms bunch length recorded was στ = 2.0 ns andhad been reduced 4 times.
Some important comments follow:(1) During the whole experiment, no collective beam instability has been observed.However, the intensity has been 5 to 8 times below the required intensity of the2.5 × 1013 bunch for the proton driver of the muon collider. The proton driverramps a batch of 4 such bunches at the cycling rate of 15 Hz. It is unclear whethercollective instability will occur or not at such a high intensity.
(2) The slip factor η is a function of momentum spread δ:
η(δ) = η0 + η1δ +O(δ2) , (2.1)
where
η0 =1
γ2t
− 1
γ2and η1 =
1
γ2
[α1 +
3β2
2
]+ η0
[α1 −
1
γ2
]. (2.2)
In above, α0 = γ−2t is the lowest-order momentum-compaction factor and α1 is the
next higher order. They are defined as
C(δ) = C0
1 + α0δ
[1 + α1δ +O(δ2)
], (2.3)
with C0 being the length of the closed orbit for the on-momentum particle and
6
-5
0
5
10
15
70 80 90 100 11
Time (ns)
FIGURE 7. Left: Beam peak current versus time after γt was dropped. The bunch was seen ex-ecuting synchrotron oscillations slowly giving a maximum peak current as a result of the shearingin momentum. The bunch shape at its narrowest instant is shown at the corner. Right: Bunchshapes before and after bunching near transition, showing a final narrow 1.5 eV-s bunch in crossesof rms width στ = 2.0 ns together with the best fitted Gaussian in solid.
C(δ) the closed orbit length at fractional momentum spread δ. Thus, even whenγ = γt, η is linear in δ and the time-slip ∆T per revolution period T0, given by
∆T
T0= ηδ = η0δ + η1δ
2 +O(δ3) , (2.4)
becomes quadratic in δ. The drift in longitudinal phase will also be quadratic inδ. Therefore, instead of shearing linearly in the momentum direction as illustratedin Fig. 3(a), the bunch will shear nonlinearly as in left plot of Fig. 8. The result isthat the final bunch will be wider. This nonlinear phase-slip can be corrected, forexample, by deploying sextupoles. It is clear that when α1 = −3
2and |η0| 1, the
first-order nonlinear drift will be eliminated.
a
a) b)
-40 0 40 -40 0 40 Time (ns)
100
E (MeV)
0
100
FIGURE 8. Effects of nonlinear η and space charge on the final phase space distribution. Leftplot shows the effect of a quadratic horizontal shear which occurs when α1 6= −3/2. Right plotshows the effect of vertical shear from strong space-charge effects.
7
(3) For an intense proton beam, space-charge effect cannot be ignored. The wakepotential is essentially proportional to the slope of the bunch linear distribution.Thus, staying near transition for too long, the bunch will shear into the shape ofthe tilted capital letter ‘N’, as shown by simulation in the right plot of Fig. 8. Thisis, in fact, a potential-well distortion of the rf wave, and can be cured, to a certainextent, by having rf systems of high frequencies, (see Sec. III B below).
III MICROWAVE INSTABILITY NEAR TRANSITION
A Analytic SolutionsIn an operation near the transition energy (η0 ≈ 0), at least the next order, η1
in Eq. (2.1), must be included for a meaningful discussion of the beam dynamics.Bogacz analyzed the stability of a coasting beam right at transition, η0 = 0 [3],by including the η1 term but neglecting other higher-order terms. For a Gaussiandistribution with rms energy spread σE, he obtained an analytic expression for thegrowth rate at the revolution harmonic n:
1
τn= −2α1nω0
(σEE0
)2
φn with tan φn =
ImZ‖0
ReZ‖0
n
, (3.5)
where ImZ‖0 > 0 implies capacitive and ω0/(2π) is the revolution frequency of the
on-energy particle which has energy E0. He drew the conclusion that the beam willbe completely stable. However, when he made this conclusion, he had in mind theassumption of α1 > 0 and φn > 0, which is not always true. As a result, there willbe microwave growth in general.
Holt and Colestock studied the same problem with coasting beam and Gaussianenergy distribution, but allowing η0 6= 0 [4]. The dispersion relation is expressed interms of the complex error function. Their conclusion is that there is no unstable
region in the complex Z‖0 -plane below transition. On the other hand, there areboth stable and unstable regions above transition. They also claimed that theirconclusion was supported by simulations. However, they did not specify the valuesof η0 and η1 in the simulations they presented or in their stability plots in the
complex Z‖0 -plane. It is hard to understand at least the situation below transition.It is clear that when |η0| is not too small, the contribution of η1 is irrelevant. Thustheir claim as stated can be interpreted as no microwave instability below transition,no matter how far away it is from transition. For this reason, this claim is quitequestionable.
When we look into the stability plots of Holt and Colestock, we can see somethingthat resembles a stability curve below transition, although the stability plots havebeen poorly drawn and are almost illegible. The presence of a stability curveimplies the existence of both stable and unstable regions, in contradiction to theirconclusion. We performed some simulations and have different results. We considera coasting beam at 100 GeV in a hypothetic ring of circumference 50 m, with arms parabolic fractional momentum spread of 0.001, interacting with a broadband
impedance of Z‖0/n = 3.00 Ω at the resonance frequency of 600 MHz and quality
8
factor Q = 1. This small size of ring is chosen because we want to limit thenumber of longitudinal bins around the ring so that not so many macro-particlewill be necessary. The Keil-Schnell circle-approximated criterion gives a limit of
|Z‖0/n| = 1.00 Ω [5]. The results are shown in Fig. 9: the top 4 plots for η = −0.005(below transition) and the lower 4 plots for η = +0.005 (above transition) at 0, 1200,2400, and 3600 turns. We see that below transition irregularities develop at the low-momentum edge and the momentum spread broadens at the low-momentum sideuntil the total spread is about 1100 MeV, about 2.75 times from the original totalspread of about 400 MeV. This definitely confirms the occurrence of microwaveinstability below transition, and the eventual self-stabilization by overshooting.Above transition, irregularities also develop at the low-momentum edge and themomentum spread also broadens at the low-momentum edge. The total spreadappears to be broader than the situation below transition. In addition, we seesmall bomb-like droplets launched at the low-momentum side, which is not observedbelow transition. We will come back to the simulations of coasting beam neartransition later in Sec. III C.
B Bunched Beam Simulations
In this section, we study the stability of a bunched beam very close to transition.As an example, take a muon bunch in the proposed 50 × 50 GeV muon collider,which has a slip factor of |η| = 1×10−6. Everything we discuss here will apply to aproton bunch also, with the exception that the muons decay while the protons arestable. We will first discuss the situation with the decay of the muons taken intoconsideration, and later push the lifetime to infinity. We assume that sextupoles andoctupoles are installed and adjusted so that the contributions of η1 and η2 becomeinsignificant compared with η0. The muon bunch we consider has an intensity ofNb = 4×1012 particles, rms width σ` = 13 cm and rms fractional momentum spreadσδ = 3× 10−5 or σE = 1.5 MeV. The impedance is assumed to be broadband with
Z‖0/n = 0.5 Ω at the angular resonant frequency of ωr = 50 GHz with quality
factor Q = 1. The muons have an e-folding lifetime of 891 turns at 50 GeV in thiscollider ring. During the muon lifetime, there is negligible phase motion. Thus abunching rf frequency system is not necessary. However, as will be explained below,rf systems are needed for the cancellation of potential-well distortion.
For bunched beams, there is the issue of potential-well distortion which must notbe mixed up with the collective microwave instability. Potential-well distortion willchange the shape of the bunch to something that looks like the right plot of Fig. 8,with the difference that the distortion of the beam does not come from the space-charge force, but mainly from the inductive part of the broadband impedance. Thewake potential seen by a particle inside a Gaussian bunch at a distance z behindthe bunch center is shown in the left plot of Fig. 10 and is given by
V (z)=e∫ z
−∞dz′ρ(z′)W0(z − z′)=− eNωrR‖
2Q cosφ0Re ejφ0−z2/(2σ2
` )w
[σ`ωrejφ0
c√
2− jz√
2σ`
],
where ρ(z) is the bunch distribution, W0(z) the longitudinal wake function, sinφ0 =
9
1/(2Q), and w is the complex error function. This distortion can be cancelled up to±3σ` by 2 rf systems [6], which at injection are at frequenciesω1/(2π) = 0.3854 GHzand ω2/(2π) = 0.7966 GHz, with voltages V1 = 65.40 kV and V2 = 24.74 kV,and phases ϕ1 = 177.20 and ϕ2 = 174.28. This compensation is shown in theleft plot of Fig. 10. Since only 2 sinusoidal rf’s are used, the cancellation is notcomplete; however, the error is less than 1% of the original wake potential and isnot important. Because of the lifetime of the muons, we first performed tracking foronly 1000 turns in the time domain using the broadband wake function W0(z). Theinitial and final bunch distributions are shown in Fig. 11. During the simulation the
FIGURE 9. The top 4 plots and lower 4 plots are for η = −0.005 (below transition) andη = +0.005 (above transition), respectively, at 0, 12000, 24000, and 36000 turns. The impedanceis a broadband with Q = 1, Z‖0/n = 3.0 Ω at the resonant frequency of 600 MHz.
10
FIGURE 10. Left: Wake potential, compensating rf voltages, and net voltage seen by particlesin the 13-cm bunch at injection. The compensating rf is the sum of two rf’s represented bydashes. Right: Wake potential seen by the simulated bunch shown as dots is interlaced with thewake potential of an ideal smooth Gaussian bunch shown in dashes. The difference (center curve)represents the random fluctuation of the finite number of macro-particles.
compensating rf voltages were lowered turn by turn to conform with the diminishingbunch intensity due to the decay of the muons.
We see from the right plot of Fig. 11 that the bunch distribution has been verymuch distorted after 1000 turns. This comes mostly from the fact that the originaldistribution of the bunch in the left plot is not exactly Gaussian. It consists of 2×106
macro-particles randomly distributed according to a bi-Gaussian distribution. As aresult, the wake potential of the actual bunch shown as a dotted curve in the rightplot of Fig. 10 deviates slightly from and wiggles around the ideal wake potentialcurve of a smooth Gaussian bunch shown in dashes. The difference is the dottedjitter curve in the center of the plot. The fluctuation seen in the right plot ofFig. 11 is the result of the accumulation of this dotted jitter curve in 1000 turnswith muon decay taken into account. Although this tiny fluctuation leads to asmall potential-well distortion in one turn (≤ 0.02 MeV), it is unfortunate that itwill accumulate turn after turn and will never reach a steady state, since the beamis so close to transition. This accumulated distortion can be computed exactlyfrom the the dotted jitter curve. Any growth in excess will come from collectivemicrowave instability. Note that the uncompensated potential-well distortion isquite different from the growth due to microwave instability. For the former, thegrowth in energy fluctuations every turn will be exactly by the same amount asgiven by the dotted jitter curve in the right Fig. 10 (if muon decay is neglected).This is because the wake potential of particles along the bunch does not dependon the energy distribution of the bunch, but only on its linear density and thelatter is essentially unchanged since the particles do not drift much during the first1000 turns. On the other hand, the initial growth due to microwave instabilityat a particular turn is proportional to the actual energy fluctuation at that turn
11
FIGURE 11. Simulation of the 13-cm bunch of 4× 1012 muons subject to a broad-band impe-dance with quality factor Q=1 and Z‖/n=0.5 Ω at the resonant angular frequency ωr=50 GHz.The half-triangular bin width is 15 ps (0.45 cm) and 2× 106 macro-particles are used. Left plotshows initial distribution with σE=1.5 MeV and σ`=13 cm. Right plot shows distribution after1000 turns with compensating rf’s depicted in Fig. 10.
and the evolution of the growth is exponential. Thus, although the growth dueto microwave instability is small at the beginning, it will be much faster later onwhen the accumulated energy fluctuations become larger. It is worth mentioningthat even if the wake potential of the initial bunch with statistical fluctuationshas been compensated exactly by the rf’s, the bunch can still be unstable againstmicrowave instability. An infinitesimal deviation from the bunch distribution canexcite the collective modes of instability corresponding to some eigenfrequencies.In other words, the accumulated growth due to potential-well distortion is a staticsolution and this static solution converges very slowly close to transition until themomentum spread is large enough for the small |η| to smooth the distribution.Microwave instability, on the other hand, is a time dependent solution.
In Fig. 12, the 3 plots on the left are for a 4000-turn simulation of the samemuon bunch using 2× 106 macro-particles with the decay of the muons considered.The two compensating rf systems are turned on. The first plot is for η = 0 so thatmicrowave instability cannot develop. All the fluctuations are due to the residualpotential-well distortion or the accumulation of the uncompensated jitters. Thesecond and third plots are for, respectively, η = −1× 10−6 (below transition) andη = +1 × 10−6 (above transition). We see that they deviate from the first plot,showing that there are growths due to microwave instability although the effect issmall. The 3 plots on the right are the same as on the left with the exception thatthe muons are considered stable, or, in other words, the particles can be protons.We see that the second and third plots differ from the first one by very much(note the change in energy scale), indicating that microwave instability does playan important role for proton bunches in a quasi-isochronous ring. We also see thatmicrowave instability is more severe above transition than below transition evenwhen the beam is so close to transition.
12
FIGURE 12. Phase-space plots of energy spread in MeV versus distance from bunch center incm at the end of 4000 turns. All are simulating 4×1014 micro-particles with 2×106 macro-particles.In the left 3 plots, the decay of the muons has been taken into account. The first left plot is forη = 0 so that it just gives the amount of potential-well distortion. The second and third plotsare for, respectively, η = −1 × 10−6 and +1 × 10−6. The small deviations from the first plot areresults of microwave instability. The right 3 plots are the same as the left, except that the muonsare considered stable. Here, large microwave growths develop (note the change of energy scale).
C Coasting Beam Simulations
For coasting beams, we do not have the inverted tilted “N”-shape wake potentialas in Fig. 10. Thus, no rf compensation will be required. However, the noise in thebeam does result in a wake potential similar to the small residual wake-potentialjitters in Fig. 10 after the rf compensation. Near transition where the phase motionis negligibly slow, these jitters will add up turn after turn without limit exactly inthe same way as the bunched beam after having optimized the rf compensation.Thus, near transition, there is essentially no difference between a coasting beam anda bunched beam after the rf compensation. The only exception is that microwaveinstability develops most rapidly near the center of the bunch where the localintensity is highest, whereas in a coasting beam, microwave instability develops
13
with equal probability along the bunch depending on the statistical fluctuations inthe macro-particles.
In Fig. 13, we show some coasting beam simulations near transition by havingη0 = 0 or ±5×10−5 and η1 = 0 or ±0.05. The coasting beam consists of 3.27×1015
protons (or nondecaying muons) having an average energy of 100 GeV in a hypo-thetic ring with circumference 50 m. The initial momentum spread is Gaussianwith rms fractional spread σδ = 0.001 or σE = 100 MeV. Thus, at 1σ, the con-tribution of |η1| = 0.05 is the same as the contribution of |η0| = 5 × 10−5. Thesimulations are performed with 8× 105 macro-particles in 400 triangular bins. The
impedance is a broadband with Q = 1 and Z‖0/n = 2 Ω at the resonant frequency
of fr = 300 MHz.
All the plots in Fig 13 are illustrated with the same scale for easy comparison.The horizontal axes are longitudinal beam position from 0 to 166.7 ns, while thevertical axes are energy spread from −4000 to 3000 MeV. Plot (a) shows the initialparticle distribution in the longitudinal phase space. All the other plots are simu-lation results at the end of 54,000 turns. Plot (b) is the result of having η0 = 0 andη1 = 0. It shows the accumulation of the wake-potential jitters over 54,000 turns.These jitters originate from the statistical fluctuation of the initial population of themacro-particles. Therefore, any deviation from Plot (b) implies microwave instabil-ity. Plots (c) and (d) are with η0 = 0, but with η1 = +0.05 and −0.05, respectively.We see the growths curl towards opposite phase directions nonlinearly as expected.This is due to the nonlinearity in δ in the time slip given by Eq. (2.4), similar tothe simulations in Fig. 8(a). It appears that Plot (c) with η1 = −0.05 gives a largergrowth. Plots (e), (g), and (i) are for η0 = −5× 10−5 (below transition), but withη1 = +0.05, −0.05, and 0, respectively. We see that the microwave instability ismost severe when η1 = 0, indicating that η1 has the ability to curb instability. Thisis, in fact, easy to understand. The phase drift driven by |η1| = 0.05 is much fasterthan that driven by |η0| = 5.0× 10−5 at larger momentum spread; for example, itwill be 4 times faster at 2σδ, 9 times faster at 3σδ, etc. As a result, a nonvanishing|η1| tends to move particles away from the clumps, thus lessening the growth dueto microwave instability.
Plots (f), (h), and (j) are for η0 = +5× 10−5 (above transition), but with η1 =+0.05, −0.05, and 0, respectively. Again microwave instability is most severe whenη1 = 0, and η1 does curb instability to a certain extent. Comparing Plots (e), (g),and (i) with Plots (f), (h), and (j), it is evident that the beam is more unstableagainst microwave instability above transition (η0 > 0) than below transition (η0 <0) independent of the sign of η1. For a fixed η0, we also notice that negative η1 ismore unstable than positive η1. The theoretical implications of these results arenontrivial and will be discussed in a future publication.
Now let us come back to the analytic investigations by Bogacz, Holt, and Cole-stock. Their results appear to contradict the simulations presented here. Analyticanalysis often starts with the Vlasov equation. The time-dependent beam distri-
14
FIGURE 13. Energy spread (MeV) versus bunch position (ns) of coasting beam simulations.See text for explanation.
15
bution ψ(φ,∆E; t) can be separated into two parts:
ψ(φ,∆E; t) = ψ0(φ,∆E) + ψ1(φ,∆E)e−iΩt . (3.6)
Here, ψ0 is the steady-state solution of the Hamiltonian and ψ1 describes the collec-tive motion of the beam with the collective frequency Ω/(2π). After linearization,the Vlasov equation becomes an eigenequation with ψ1 as the eigenfunction andΩ/(2π) the eigenfrequency. The equation also depends on ψ0. Thus we must solvefor the steady-state solution first before solving the eigenequation. The steady-statesolution is the time-independent solution of the Hamiltonian which includes the con-tribution of the wake function. In other words, ψ0 is the potential-well-distortedsolution. Far away from transition, this distortion is mostly in the φ coordinate, forexample, those brought about by the space-charge or inductive forces. Therefore,for a coasting beam, there will not be any potential-well distortion at all. The sit-uation, however, is quite different close to transition. As was pointed out in above,the potential-well distortion is now in the ∆E coordinate. For this reason, notonly bunched beams, even coasting beams will suffer from potential-well distortionas a result of the nonuniformity of the beam. In simulations, the nonuniformityarrives from the statistical fluctuation of the distribution of the macro-particles.This nonuniformity will accumulate turn by turn until the momentum spread is solarge that the small |η| is able to smooth out all nonuniformity. In other words, thesteady-state distribution ψ0 that goes into the Vlasov equation will be completelydifferent from the original distribution in the absence of the wake. In the analysisof Bogacz, Holt, and Colestock, the ideal smooth Gaussian distribution in energywas substituted for ψ0 in the Vlasov equation. However, this is a very unstablestatic distribution; even a small perturbation will accumulate turn by turn withextremely slow convergence. For this reason, it is hard to understand what theirresults really represent.
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